Electrothermal feedback in superconducting nanowire single-photon ...

Electrothermal feedback in superconducting nanowire single-photon detectors Andrew J. Kerman,1 Joel K.W. Yang,2 Richard J. Molnar,1 Eric A. Dauler,1, 2 and Karl K. Berggren2

arXiv:0812.0290v1 [cond-mat.supr-con] 1 Dec 2008

2

1 Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, MA, 02420 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, 02139 (Dated: December 1, 2008)

We investigate the role of electrothermal feedback in the operation of superconducting nanowire single-photon detectors (SNSPDs). It is found that the desired mode of operation for SNSPDs is only achieved if this feedback is unstable, which happens naturally through the slow electrical response associated with their relatively large kinetic inductance. If this response is sped up in an effort to increase the device count rate, the electrothermal feedback becomes stable and results in an effect known as latching, where the device is locked in a resistive state and can no longer detect photons. We present a set of experiments which elucidate this effect, and a simple model which quantitatively explains the results. PACS numbers: 74.76.Db, 85.25.-j

Superconducting nanowire single-photon detectors (SNSPDs) [1, 2, 3, 4] have a combination of attributes found in no other photon counter, including high speed, high detection efficiency over a wide range of wavelengths, and low dark counts. Of particular importance is their high single-photon timing resolution of ∼30 ps [4], which permits extremely high data rates in communications applications [5, 6]. Full use of this electrical bandwidth is limited, however, by the fact that the maximum count rates of these devices are much smaller (a few hundred MHz for 10 µm2 active area, and decreasing as the area is increased [2]), limited by their large kinetic inductance and the input impedance of the readout circuit [2, 7]. To increase the count rate, therefore, one must either reduce the kinetic inductance (by using a smaller active area or through the use of different materials or substrates) or increase the load impedance [7]. However, either one of these approaches causes the wire to “latch” into a stable resistive state where it no longer detects photons [8]. This effect arises when negative electrothermal feedback, which in normal operation allows the device to reset itself, is made fast enough that it becomes stable. We present experiments which probe the stability of this feedback, and we develop a model which quantitatively explains our observations. The operation of an SNSPD is illustrated in Fig. 1(a). A nanowire (typically ∼100nm wide, 5nm thick) is biased with a DC current I0 near the critical current Ic . When a photon is absorbed, a short ( 0), if less it will contract (vns < 0). We can use eq. 1 to describe the electrothermal circuit in fig. 1(a), by combining it with: dRn /dt = 2ρn vns and Id Rn + LdId /dt = RL (I0 − Id ). Assuming that the small-signal stability of the DC hotspot solution (Id = Iss , Rss = RL (I0 /Iss − 1)) determines whether it will

2 latch, we linearize about this solution to obtain apsecondorder system, with damping coefficient ζ = 4II0ss τth /τe , where τth ≡ RL /2ρn v0 is a thermal time constant. This p can be re-expressed as ζ = 14 τth,tot /τe,tot , with τe,tot ≡ L/Rtot and τth,tot ≡ Rtot /2ρn v0 , where Rtot ≡ RL + Rss . If the damping is less than some critical, minimum value ζlatch , the feedback cannot stabilize the hostpot during the initial photoresponse, as described above, and the device operates normally. However, since the steadystate solution gives Rss ∝ I0 (as I0 is increased, a larger hotspot is necessary for Id = Iss ) the hotspot becomes more stable as I0 is increased, until eventually ζ(Ilatch ) = ζlatch and the device latches. For a correctly functioning device, Ilatch > Ic , so that latching does not affect its operation. However, if τe is decreased, Ilatch decreases, and eventually it becomes less than Ic . This prevents the device from being biased near Ic [17], resulting in a drastic reduction in performance [12, 13]. Devices used in this work were fabricated from ∼5 nm thick NbN films, deposited on R-plane sapphire substrates in a UHV DC magnetron sputtering system (base pressure < 10−10 mbar). Film deposition was performed at a wafer temperature of ∼800 C and a pressure of ∼ 10−8 mbar [15]. Aligned photolithography and liftoff were used to pattern ∼100 nm thick Ti films for on-chip resistors [8], and Ti:Au contact pads. Patterning of the NbN was then performed with e-beam lithography using HSQ resist [3]. Devices were tested in a cryogenic probing station at temperatures of 1.8-12K using the techniques described in Refs. [2, 3]. Figure 1(c)-(f) show data for a set of (3µm×3.3µm active area) devices having various resistors RS in series with the 50Ω readout line [8] [fig. 1(b)] so that RL = 50Ω+RS . Panels (c) and (d) show averaged pulse shapes for devices with RS = 0 and RS = 250Ω, respectively. Clearly, the reset time can be reduced; however, this comes at a price. Panels (e) and (f) show, for devices with different RS , the current Iswitch ≡ min(Ic , Ilatch ) above which each device no longer detects photons, and the measured detection efficiency (DE) at I0 = 0.975Iswitch [17]. The data are plotted vs. the speedup of the reset time (RS + 50Ω)/50Ω, and show that as this speedup is increased, Iswitch decreases (due to reduction of Ilatch ), resulting in a significantly reduced DE. To investigate the latched state, we fabricated devices designed to probe the stability of self-heating hotspots as a function of I0 , L, and RL . Each device consisted of three sections in series, as shown in Fig. 2 (a): a 3 µmlong, 100 nm-wide nanowire where the hotspot was nucleated [11]; a wider (200 nm) meandered section acting as an inductance; and a series of nine contact pads interspersed with Ti-film resistors. Also shown are the two electrical probes, which result in the circuit of Fig. 2(b): a high-impedance (Rp = 20kΩ) 3-point measurement of the nanowire resistance. We varied RL by touching the probes down to different pads along the line, and L by testing different devices (with different series inductors). We tested 66 devices on three chips, and selected from

Detector

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Rn Id

Detector

e)

d)

f)

RL

I0

L

b)

c)

RS

bias tee

Rn Id L

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FIG. 1: Figure 1:(color online) Speedup and latching of nanowire detectors with increased load impedance. (a) electrical model of detector operation. A hotspot is nucleated by absorption of a photon, producing a resistance Rn in series with the wire’s kinetic inductance L. (b) circuit describing experimental configuration for increasing load resistance using a series resistor RS . (c) and (d) averaged pulse shapes from 3µm×3.3µm detectors with L ∼ 50 nH for RS = 0 and RS = 250Ω; red solid lines are predictions with no free parameters. (e) switching current Iswitch ≡ min(Ic , Ilatch ) vs. speedup 50Ω/(RS + 50Ω). For small RS , Iswitch = Ic but as RS is increased, Ilatch decreases, becoming less than Ic . (f) DE at I0 = 0.975Iswitch vs. speedup (open squares). Also shown (crosses) are the expected DEs assuming that latching affects DE simply by limiting I0 (obtained from DE vs. I0 at RS = 0).

these only unconstricted [12] wires with nearly identical linewidths (the observed Ic of devices used here were within ∼10% of each other - typically 22-24 µA), spanning the range: RL = 20 − 1000Ω and L = 6 − 600 nH. For each L and RL , we acquired a DC I-V curve like those shown in Figs. 2(c),(d), sweeping I0 downward starting from high values where the hotspot was stable [14]. All curves exhibit a step in voltage which can be identified as Ilatch . For smaller RL , where Ilatch is large, Ic can be identified as the point where an onset of resistance occurs common to several curves; this onset is more gradual with I0 rather than sudden as at Ilatch . These features can be understood by examining figs. 2(e) and (f), which show Id and Rn inferred from the data of fig. 2(d). Above the discontinuous jump in current at Ilatch , Id is fixed at Iss (independent of I0 and RL ) indicating the latched state. For smaller τe , Ilatch < Ic , so only Ilatch is observed [18]. When τe is large enough that Ilatch > Ic , an intermediate region appears where the resistance increases continuously with I0 ; this arises from relaxation oscillations [9, 16], as indicated in the figure: the device cannot superconduct when I0 > Ic , but neither can a stable hotspot be formed when I0 < Ilatch , so instead current oscillates back and forth between the device and the load, producing a periodic pulse train with a frequency that increases as I0 is increased [17]. The average resistance increases with this frequency, producing the observed continuous decrease in Id . Figure 3 shows the measured Ilatch as a function of RL and L, which can be thought of as defining the boundary

3 a)

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L=605 nH

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RL=333Ω Ω 33Ω Ω

34Ω Ω

217nH

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stable

unstable

323Ω Ω

L=6.0 nH 34Ω Ω

34Ω Ω

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FIG. 2: Figure 2: (color online) Hotspot stability measurements. (a) device schematic; two ground-signal-ground probes simultaneously perform a high-impedance 3-point measurement of the hotspot resistance, with RL determined by probe position. (b) equivalent electrical circuit. (c) and (d) example V-I curves, with L = 60 nH and L = 605 nH, respectively; (e) and (f) inferred Id and Rn /RL for the data shown in (d). In (e) the relaxation oscillations in the region Ic < I0 < Ilatch are shown schematically. Dashed lines show (e) Id = I0 and (f) Rss = RL (I0 /Iss − 1).

between stable and unstable hotspots. Our simple model described above predicts: τe /τth ∝ (Ilatch /Iss )2 , a line of slope 1 in the figure (indicated by the dashed line). The data do approach this line, though only in the τe ≫ τth limit. This is consistent with the assumption of constant (or slowly varying) Id under which eq. 1 was derived. As τe /τth is decreased, the data trend downward, away from this line, and Ilatch /Iss becomes almost independent of τe /τth ; this implies a minimum Ilatch /Iss , or equivalently, a minimum Rss /RL , below which the hotpost is always unstable. This is shown in the inset: the measured minimum stable Rss is always greater than ∼ RL . The crossover to this behavior can be explained in terms of a timescale τa over which the temperature profile of the hotspot stabilizes into the steady-state form which yields eq. 1. For power density variations occuring faster than this, the NS boundaries do not have time to start moving, resulting instead in a temperature deviation ∆T . Since the NS boundary occurs at T ≈ Tc , where ρn is temperature-dependent (defined by dρ/dT ≡ β > 0), this changes Rn , giving a second, parallel electrothermal feedback path which dominates for frequencies ω ≫ τa−1 . We can describe this by replacing eq. 1 with: d2 l dl γρn 2 τa 2 + = Id2 ρ(∆T ) − Iss ρn 2 dt dt

(2)

γρn τa d2 l d∆T − h∆T = dt 2 dt2

(3)

c

Here, l is the hotspot length, ρ(∆T ) is the resistance per unit length, and Rn = ρ(∆T )l. In eq. 2, τa is the charac-

FIG. 3: Figure 3: (color online) Summary of hotspot stability results. Data are shown from 3 different chips (indicated by different colors). Circles, squares, and triangles are data for L =6-12, 15-60, and 120-600 nH, respectively. In the τe ≫ τa limit (where NS domain-wall motion dominates the electrothermal feedback), the data approach the dashed line, which is the prediction based on eq. 1, discussed in the text. For τe ≪ τth the NS domain walls are effectively fixed and the temperature feedback dominates. In this regime the feedback is always unstable when Rss ∼> RL (or equivalently, I0 ∼< 2Iss ), as shown in the inset. The solid curves are obtained from eq. 8 assuming a phase margin of 30 degrees; each curve corresponds to a fixed L in the set (6,15,30,60,600) nH and spans the range of RL in the data. The dotted lines extend these predictions over a wider range of RL .

teristic time over which dl/dt = 2vns adapts to changes in power density: for slow timescales dt ≫ τa , τa d2 l/dt2 ≪ dl/dt and eq. 2 reduces to eq. 1 (with ρ = ρn ). For faster timescales, τa d2 l/dt2 becomes appreciable, and acts as a source term for temperature deviations in eq. 3. When dt ≪ τa , τa d2 l/dt2 ≫ dl/dt and eqs. 2 and 3 can be com2 ρn − h∆T [19]. In this bined to give: cd∆T /dt ≈ Id2 ρ− Iss limit, if RL ∼ Rss the bias circuit including RL begins to look like a current source, which then results in positive feedback: a current change produces a temperature and resistance change of the same sign. Therefore, the hotspot is always unstable when Rss < RL . Expressing eqs. 2 and 3 in dimensionless units (i ≡ Id /I0 , r ≡ Rn /RL , λ ≡ lβTc /RL , θ ≡ T /Tc), and expanding to first order in small deviations from steady state, we obtain: δi′ = − i0 δi + i−1 0 δr

(4)

δr = η (i0 − 1) δθ + η −1 δλ

(5)

τa ′′ τe δλ + δλ′ = 2η 2 δθ + 2i0 η −1 δi τe τth

(6)

δθ′ =

Θτth τa ′′ τe δλ − δθ ητe τc τc

(7)

4 Here, the prime denotes differentiation with respect to t/τe , i0 ≡ I0 /Iss , Θ ≡ (Tc − T0 )/Tc , η ≡ βTc /ρn characterizes the resistive transition slope, and τc ≡ c/h is a cooling time constant. When τe ≫ τth , τa , the system reduces to: δi′′ + i0 δi′ − p4τe /τth −1≈ 0, which has damping coefficient ζ = i0 (4 τe /τth ) , as above. In the opposite limit, where τe ≪ τth , τa , we obtain: δi′′ + i0 δi′ + (2ηΘτe /τc )(i0 − 2) ≈ 0. In agreement with our argument above, the oscillation frequency becomes negative for Rss < RL (I0 < 2Iss ). We characterize the stability of the system of eqs. 47 using its “open loop” gain Aol : we assume a small oscillatory perturbation by replacing δr in eq. 4 with ∆rejωt , and responses (δi, δθ, δλ, δr)ejωt . Solving for Aol ≡ δr/∆r, we obtain: Aol =

e (1 + jω ττec ) − 4ηΘω 2 (i0 − 1) ττae 4 ττth h i τa τc τa 2jωηΘ ) − (1 + jω )(1 + jω ) jωi0 (1 + jω i0 τe τe τe

(8) The stability of the system can then be quantified by the phase margin: π+arg[Aol (ω0 )], where ω0 is the unity gain (|Aol | = 1) frequency. In the extreme case, when the phase margin is zero (arg[Aol (ω0 )] = −π), the feedback is positive. The solid lines in Fig. 3 show our best fit to the data. Note that although the stability is determined only by τe /τth and i0 in the two extreme limits (not visible in the figure), in the intermediate region of interest here this is not the case, so several curves are shown. Each solid curve segment corresponds to a single L, over the range of RL tested; the dotted lines continue these curves for a wider range of RL . The data are grouped into

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three inductance ranges: 6-12, 15-60, and 120-600 nH, indicated by circles, squares, and triangles, respectively. We used fixed values Θ = 0.8, η = 6.5, which are based on independent measurements, and fitted τa = 1.9 ns, and τc = 7.7 ns to all data. Separate values of ρn v0 were fitted to data from each of the three chips, differing at most by a factor of ∼2. These fitted values were ρn v0 ∼ 1 × 1011 Ω/s; since ρn ∼ 109 Ω/m, this gives v0 ∼ 100 m/s, a reasonable value. A natural question to ask in light of this analysis is whether it suggests a method for speeding up these devices. The most obvious way would be to increase the heat transfer coefficient h, which increases both Iss and v0 , moving the wire further into the unstable region, and allowing its speed to be increased further without latching. However, at present it is unknown how much h can be increased before the DE begins to suffer. At some point, the photon-generated hotspot will disappear too quickly for the wire to respond in the desired fashion. In any case, experiments like those described here will be a useful measurement tool in future work for understanding the impact of changes in the material and/or substrate on the thermal coupling and electrothermal feedback. We acknowledge helpful discussions with Sae Woo Nam, Aaron Miller, Enectal´ı Figueroa-Feliciano, and Jeremy Sage. This work is sponsored by the United States Air Force under Contract #FA8721-05-C-0002. Opinions, interpretations, recommendations and conclusions are those of the authors and are not necessarily endorsed by the United States Government.

Berggren, Opt. Express. 14, pp. 527-534 (2006). [4] E.A. Dauler, A.J. Kerman, B.S. Robinson, J.K.W. Yang, B. Voronov, G. Gol’tsman, S.A. Hamilton, and K.K. Berggren, e-print physics/0805.2397. [5] D. Rosenberg, S.W. Nam, P.A. Hiskett, C.G. Peterson, R.J. Hughes, J.E. Nordholt, A.E. Lita, A.J. Miller, Appl. Phys. Lett 88, p. 21108, (2006); H. Takesue, S.W. Nam, Q. Zhang, R.H. Hadfield, T. Honjo, K. Tamaki, Y. Yamamoto, Nature Photonics 1, p. 343 (2007). [6] B.S. Robinson, A.J. Kerman, E.A. Dauler, D.M. Boroson, S.A. Hamilton, J.K.W. Yang, V. Anant, and K.K. Berggren Proc. SPIE 6709, p. 67090Z (2007). [7] An alternative has recently been demonstrated involving many parallel nanowires [A. Korneev, A. Divochiy, M. Tarkhov, O. Minaeva, V. Seleznev, N. Kaurova, B. Voronov, O. Okunev, G. Chulkova, I. Milostnaya, K. Smirnov, and G. Goltsman, J. Phys. Conf. Series 97 p. 012307 (2008)], however, it is as yet unknown whether this method can also achieve the high detection efficiency and low jitter of state-of-the art conventional devices. [8] J.K.W. Yang, A.J. Kerman, E.A. Dauler, V. Anant, K.M. Rosfjord, and K.K. Berggren IEEE Trans. Appl. Supercond. 17, p. 581 (2007). [9] A.Vl. Gurevich and R.G. Mints, Rev. Mod. Phys. 59,

5 p.941 (1987), and references therein. [10] This description is further simplified by the fact that near the NS boundary all material properties can be approximated by their values at Tc . [11] The sheet resistance of the NbN is R ≈ 500Ω, and that the longest hotspot measured in this work is ∼300 nm. [12] A.J. Kerman, E.A. Dauler, J.K.W. Yang, K.M. Rosfjord, V. Anant, K.K. Berggren, G.N. Gol’tsman, and B.M. Voronov, Appl. Phys. Lett. 90, p. 101110 (2007). [13] We have focused on the impact of electrothermal feedback on reset time, but it may also influence the timing jitter since it opposes the fast, initial growth of the normal domain that produces the sharp leading-edge of the output pulses. [14] The results are almost identical when sweeping I0 upward, since the dark counts of the device allow it to lock into the latched state if it is stable. [15] R.J. Molnar, E.A. Dauler, A.J. Kerman, and K.K.

Berggren, to be published. [16] R.H. Hadfield, A.J. Miller, S.W. Nam, R.L. Kautz, and R.E. Schwall, Appl. Phys. Lett. 87, p. 203505 (2005). [17] In typical experiments, the DC impedance to ground RDC is determined by the I0 source behind a bias tee, [Fig. 1(b) with RS = 0]. When Ic is exceeded, the detector oscillates (if Ilatch > Ic ), for the time constant of the bias tee, after which it senses the larger RDC and latches. The absence of this burst of pulses is a signature for Ilatch < Ic . [18] When Ilatch < Ic , Ilatch is equivalent to what is often called the “retrapping” current in superconducting devices exhibiting self-heating. [19] This situation is identical to that encountered in superconducting transition-edge sensors (see, e.g.: K.D. Irwin, G.C. Hilton, D.A. Wollman, and J.M. Martinis, J. Appl. Phys. 83, p.3978 (1998)).